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Theorem finlocfin 23379
Description: A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypotheses
Ref Expression
finlocfin.1 𝑋 = βˆͺ 𝐽
finlocfin.2 π‘Œ = βˆͺ 𝐴
Assertion
Ref Expression
finlocfin ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) β†’ 𝐴 ∈ (LocFinβ€˜π½))

Proof of Theorem finlocfin
Dummy variables 𝑛 𝑠 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1133 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) β†’ 𝐽 ∈ Top)
2 simp3 1135 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) β†’ 𝑋 = π‘Œ)
3 simpl1 1188 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐽 ∈ Top)
4 finlocfin.1 . . . . . 6 𝑋 = βˆͺ 𝐽
54topopn 22763 . . . . 5 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
63, 5syl 17 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝑋 ∈ 𝐽)
7 simpr 484 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
8 simpl2 1189 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ Fin)
9 ssrab2 4072 . . . . 5 {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) β‰  βˆ…} βŠ† 𝐴
10 ssfi 9175 . . . . 5 ((𝐴 ∈ Fin ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) β‰  βˆ…} βŠ† 𝐴) β†’ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) β‰  βˆ…} ∈ Fin)
118, 9, 10sylancl 585 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) β‰  βˆ…} ∈ Fin)
12 eleq2 2816 . . . . . 6 (𝑛 = 𝑋 β†’ (π‘₯ ∈ 𝑛 ↔ π‘₯ ∈ 𝑋))
13 ineq2 4201 . . . . . . . . 9 (𝑛 = 𝑋 β†’ (𝑠 ∩ 𝑛) = (𝑠 ∩ 𝑋))
1413neeq1d 2994 . . . . . . . 8 (𝑛 = 𝑋 β†’ ((𝑠 ∩ 𝑛) β‰  βˆ… ↔ (𝑠 ∩ 𝑋) β‰  βˆ…))
1514rabbidv 3434 . . . . . . 7 (𝑛 = 𝑋 β†’ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) β‰  βˆ…})
1615eleq1d 2812 . . . . . 6 (𝑛 = 𝑋 β†’ ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) β‰  βˆ…} ∈ Fin))
1712, 16anbi12d 630 . . . . 5 (𝑛 = 𝑋 β†’ ((π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) ↔ (π‘₯ ∈ 𝑋 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) β‰  βˆ…} ∈ Fin)))
1817rspcev 3606 . . . 4 ((𝑋 ∈ 𝐽 ∧ (π‘₯ ∈ 𝑋 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) β‰  βˆ…} ∈ Fin)) β†’ βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
196, 7, 11, 18syl12anc 834 . . 3 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
2019ralrimiva 3140 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
21 finlocfin.2 . . 3 π‘Œ = βˆͺ 𝐴
224, 21islocfin 23376 . 2 (𝐴 ∈ (LocFinβ€˜π½) ↔ (𝐽 ∈ Top ∧ 𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
231, 2, 20, 22syl3anbrc 1340 1 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = π‘Œ) β†’ 𝐴 ∈ (LocFinβ€˜π½))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  {crab 3426   ∩ cin 3942   βŠ† wss 3943  βˆ…c0 4317  βˆͺ cuni 4902  β€˜cfv 6537  Fincfn 8941  Topctop 22750  LocFinclocfin 23363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7853  df-1o 8467  df-en 8942  df-fin 8945  df-top 22751  df-locfin 23366
This theorem is referenced by:  locfincmp  23385  cmppcmp  33368
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